Define as , and . Prove by induction that
Let P(n) be the propositional statement for n>1 " "
The LHS of P(2) is 5^2-2^2=21 and the RHS if P(2) is 7(3)-10(0)=21. So P(2) is true.
Now assume that P(j) is true for all j<k. That is assume . Then by this assumption that P(j-1) can be written as
Now I'm kind of stuck. This is either going to take some sick algebra or I made a boo-boo.
EDIT: On second thought, maybe just writing that in terms of j+1 would be better? I'm not 100% sure. I get the proof. My teacher is just a major nitpicker with tiny details.
You're absolutely right. The text we have for some reason makes a case for two types of induction, the first one as you just described and the second one like this:
"Suppose for each number, n, P(n) is a statement associated with n and the statement has the following properties:
a)If P(k) is true whenever P(j) holds for all natural numbers j<k, then P(n) holds for all natural numbers n."
This is essentially the same method, but for some reason when we reference the induction hypothesis we must pick one of the two. My teacher is kind of strange. But I got the proof now. Thank you very much!